In sharp contrast with balanced OT, UOT Sinkhorn algorithm might converge slowly, even if \(\epsilon \)  is large.
For instance when marginals are penalized with the Kullback-Leibler divergence \(\phi =\rho \) , Sinkhorn converges linearly at a rate \((1+\tfrac{\epsilon }{\rho })^{-1}\)  for the sup-norm \({\cdot }_\infty \)  [1]}.
This rate is close to 1 and thus the convergence is slow when \(\epsilon \ll \rho \) .
This issue motivates in [2]} the introduction of a translation invariant formulation of the dual problem with associated Sinkhorn-like iterations enjoying better convergence rates.
